Dichotomy of global capacity density in metric measure spaces

Hiroaki Aikawa; Anders Björn; Jana Björn; Nageswari Shanmugalingam

doi:10.1515/acv-2016-0066

Article

Dichotomy of global capacity density in metric measure spaces

Hiroaki Aikawa , Anders Björn , Jana Björn and Nageswari Shanmugalingam

Published/Copyright: May 16, 2017

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From the journal Advances in Calculus of Variations Volume 11 Issue 4

Abstract

The variational capacity capp in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E⊂ℝn,

infx∈ℝn⁡capp⁡(E∩B⁢(x,r),B⁢(x,2⁢r))capp⁡(B⁢(x,r),B⁢(x,2⁢r))

is either zero or tends to 1 as r→∞. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in ℝn. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

Keywords: Capacitarily stable collection; capacitary potential; capacity density; dichotomy; metric space; Sobolev capacity; variational capacity

MSC 2010: 31C15; 31C45; 31E05

Communicated by Frank Duzaar

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: JP25287015

Award Identifier / Grant number: JP25610017

Award Identifier / Grant number: JP17H01092

Funding source: Vetenskapsrådet

Award Identifier / Grant number: 621-2011-3139

Award Identifier / Grant number: 621-2014-3974

Award Identifier / Grant number: 2016-03424

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1200915

Award Identifier / Grant number: DMS-1500440

Funding statement: The first author was partially supported by JSPS KAKENHI grants JP25287015, JP25610017 and JP17H01092. The second and third authors were partially supported by the Swedish Research Council grants 621-2011-3139, 621-2014-3974 and 2016-03424. The last author was partially supported by NSF grants DMS-1200915 and DMS-1500440. Part of this research was conducted during the last author’s visit to Linköping University; she wishes to thank that institution for its kind hospitality.

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Received: 2016-12-29

Revised: 2017-04-24

Accepted: 2017-04-27

Published Online: 2017-05-16

Published in Print: 2018-10-01

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https://doi.org/10.1515/acv-2016-0066

Keywords for this article

Capacitarily stable collection; capacitary potential; capacity density; dichotomy; metric space; Sobolev capacity; variational capacity